Properties

Label 441.2.a.f
Level 441
Weight 2
Character orbit 441.a
Self dual Yes
Analytic conductor 3.521
Analytic rank 1
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 441.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(3.52140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{4} - 2q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + 2q^{20} - 4q^{22} - q^{25} + 2q^{26} + 2q^{29} + 5q^{32} - 6q^{34} + 6q^{37} - 4q^{38} + 6q^{40} + 2q^{41} - 4q^{43} + 4q^{44} - q^{50} - 2q^{52} - 6q^{53} + 8q^{55} + 2q^{58} + 12q^{59} + 2q^{61} + 7q^{64} - 4q^{65} + 4q^{67} + 6q^{68} + 6q^{73} + 6q^{74} + 4q^{76} - 16q^{79} + 2q^{80} + 2q^{82} - 12q^{83} + 12q^{85} - 4q^{86} + 12q^{88} - 14q^{89} + 8q^{95} - 18q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 0 −1.00000 −2.00000 0 0 −3.00000 0 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2} - 1 \)
\( T_{5} + 2 \)
\( T_{13} - 2 \)